I don't think that there is a "standard terminology" for a ring $R$ whose left zero divisors are also right zero divisors. The "reversible rings" that P.M.Cohn considers here are rings satisfying the stronger condition $ab=0 \Rightarrow ba=0$. Usually in the theory of Ore localization, you would call a multiplicative subset $S$ of a ring $R$, "right reversible" if $sr=0 \Rightarrow \exists s'\in S: rs'=0$ for any $r\in R$ and $s\in S$ (see for instance here). Hence what you want is that $R$ is a right reversible set in the sense of Ore localization, but this would somehow clash with Cohn's terminology.

In my opinion the best would be not to introduce another terminology. Saying that you are considering a ring whose left zero divisors are also right zero divisors is self-explaining.

I don't think that there is a "standard terminology" for a ring $R$ whose left zero divisors are also right zero divisors. The "reversible rings" that P.M.Cohn considers here are rings satisfying the stronger condition $ab=0 \Rightarrow ba=0$. Usually in the theory of Ore localization, you would call a multiplicative subset $S$ of a ring $R$, "left reversible" if $rs=0 \Rightarrow \exists s'\in S: s'r=0$ for any $r\in R$ and $s\in S$ (see for instance here). Hence what you want is that $S=R\setminus\{0\}$ is a left reversible set in the sense of Ore localization, but this would somehow clash with Cohn's terminology.

In my opinion the best would be not to introduce another terminology. Saying that you are considering a ring whose left zero divisors are also right zero divisors is self-explaining.